An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order n forms a unital algebra since the identity matrix of order n is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a (unital) ring that is also a vector space.

Many authors use the term algebra to mean associative algebra, or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra.

Definition and motivation

First example: The complex numbers

Any complex number may be written a + bi, where a and b are real numbers and i is the imaginary unit. In other words, a complex number is represented by the vector (a, b) over the field of real numbers. So the complex numbers form a two-dimensional real vector space, where addition is given by (a, b) + (c, d) = (a + c, b + d) and scalar multiplication is given by c(a, b) = (ca, cb), where all of a, b, c and d are real numbers. We use the symbol · to multiply two vectors together, which we use complex multiplication to define: (a, b) · (c, d) = (ac − bd, ad + bc).

The following statements are basic properties of the complex numbers. If x, y, z are complex numbers and a,b are real numbers, then

(x + y) · z = (x · z) + (y · z). In other words, multiplying a complex number by the sum of two other complex numbers, is the same as multiplying by each number in the sum, and then adding.

(ax) · (by) = (ab) (x · y). This shows that complex multiplication is compatible with the scalar multiplication by the real numbers.

This example fits into the following definition by taking the field K to be the real numbers, and the vector space A to be the complex numbers.

Definition

Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · (i.e. if x and y are any two elements of A, x · y is the product of x and y). Then A is an algebra over K if the following identities hold for all elements x, y, and z of A, and all elements (often called scalars) a and b of K:

These three axioms are another way of saying that the binary operation is bilinear. An algebra over K is sometimes also called a K-algebra, and K is called the base field of A. The binary operation is often referred to as multiplication in A. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily associative, although some authors use the term algebra to refer to an associative algebra.

Notice that when a binary operation on a vector space is commutative, as in the above example of the complex numbers, it is left distributive exactly when it is right distributive. But in general, for non-commutative operations (such as the next example of the quaternions), they are not equivalent, and therefore require separate axioms.

A motivating example: quaternions

The real numbers may be viewed as a one-dimensional vector space with a compatible multiplication, and hence a one-dimensional algebra over itself. Likewise, as we saw above, the complex numbers form a two-dimensional vector space over the field of real numbers, and hence form a two dimensional algebra over the reals. In both these examples, every non-zero vector has an inverse, making them both division algebras.
Although there are no division algebras in 3 dimensions, in 1843, the quaternions were defined and provided the now famous 4-dimensional example of an algebra over the real numbers, where one can not only multiply vectors, but also divide. Any quaternion may be written as (a, b, c, d) = a + bi + cj + dk. Unlike the complex numbers, the quaternions are an example of a non-commutative algebra: for instance, (0,1,0,0) · (0,0,1,0) = (0,0,0,1) but (0,0,1,0) · (0,1,0,0) = (0,0,0,−1).

The quaternions were soon followed by several other hypercomplex number systems, which were the early examples of algebras over a field.

Basic concepts

Algebra homomorphisms

Given K-algebras A and B, a K-algebra homomorphism is a K-linear mapf: A → B such that f(xy) = f(x) f(y) for all x,y in A. The space of all K-algebra homomorphisms between A and B is frequently written as

HomK-alg(A,B).{\displaystyle \mathbf {Hom} _{K{\text{-alg}}}(A,B).}

A K-algebra isomorphism is a bijectiveK-algebra homomorphism. For all practical purposes, isomorphic algebras differ only by notation.

Subalgebras and ideals

A subalgebra of an algebra over a field K is a linear subspace that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset L of a K-algebra A is a subalgebra if for every x, y in L and c in K, we have that x · y, x + y, and cx are all in L.

In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.

A left ideal of a K-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset L of a K-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following three statements.

1) x + y is in L (L is closed under addition),

2) cx is in L (L is closed under scalar multiplication),

3) z · x is in L (L is closed under left multiplication by arbitrary elements).

If (3) were replaced with x · z is in L, then this would define a right ideal. A two-sided ideal is a subset that is both a left and a right ideal. The term ideal on its own is usually taken to mean a two-sided ideal. Of course when the algebra is commutative, then all of these notions of ideal are equivalent. Notice that conditions (1) and (2) together are equivalent to L being a linear subspace of A. It follows from condition (3) that every left or right ideal is a subalgebra.

It is important to notice that this definition is different from the definition of an ideal of a ring, in that here we require the condition (2). Of course if the algebra is unital, then condition (3) implies condition (2).

Extension of scalars

If we have a field extensionF/K, which is to say a bigger field F that contains K, then there is a natural way to construct an algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product VF:=V⊗KF{\displaystyle V_{F}:=V\otimes _{K}F}. So if A is an algebra over K, then AF{\displaystyle A_{F}} is an algebra over F.

Kinds of algebras and examples

Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different.

Unital algebra

An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.

Zero algebra

An algebra is called zero algebra if uv = 0 for all u, v in the algebra,[2] not to be confused with the algebra with one element. It is inherently non-unital (except in the case of only one element), associative and commutative.

One may define a unital zero algebra by taking the direct sum of modules of a field (or more generally a ring) K and a K-vector space (or module) V, and defining the product of every pair of elements of V to be zero. That is, if λ, μ ∈ K and u, v ∈ V, then (λ + u) (μ + v) = λμ + (λv + μu). If e1, ... ed is a basis of V, the unital zero algebra is the quotient of the polynomial ring K[E1, ..., En] by the ideal generated by the EiEj for every pair (i, j).

An example of unital zero algebra is the algebra of dual numbers, the unital zero R-algebra built from a one dimensional real vector space.

These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or modules. For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring R = K[x1, ..., xn] over a field. The construction of the unital zero algebra over a free R-module allows extending this theory as a Gröbner basis theory for sub modules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.

Non-associative algebra

A non-associative algebra[3] (or distributive algebra) over a field K is a K-vector space A equipped with a K-bilinear mapA×A→A{\displaystyle A\times A\rightarrow A}. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited. That is, it means "not necessarily associative".

Algebras and rings

The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a ringA together with a ring homomorphism

η:K→Z(A),{\displaystyle \eta \colon K\to Z(A),}

where Z(A) is the center of A. Since η is a ring morphism, then one must have either that A is the zero ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication

K×A→A{\displaystyle K\times A\to A}

given by

(k,a)↦η(k)a.{\displaystyle (k,a)\mapsto \eta (k)a.}

Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as

f(ka)=kf(a){\displaystyle f(ka)=kf(a)}

for all k∈K{\displaystyle k\in K} and a∈A{\displaystyle a\in A}. In other words, the following diagram commutes:

Structure coefficients

For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A.
Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i.e., so the resulting multiplication satisfies the algebra laws.

Thus, given the field K, any finite-dimensional algebra can be specified up toisomorphism by giving its dimension (say n), and specifying n3structure coefficientsci,j,k, which are scalars.
These structure coefficients determine the multiplication in A via the following rule:

Note however that several different sets of structure coefficients can give rise to isomorphic algebras.

In mathematical physics, the structure coefficients are generally written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, the structure coefficients are often written ci,jk, and their defining rule is written using the Einstein notation as

eiej = ci,jkek.

If you apply this to vectors written in index notation, then this becomes

(xy)k = ci,jkxiyj.

If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.

Classification of low-dimensional algebras

Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.[4]

There exist two two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and a. According to the definition of an identity element,

There exist five three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), a and b. Taking into account the definition of an identity element, it is sufficient to specify

aa=a,bb=b,ab=ba=0{\displaystyle \textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0} for the first algebra,

aa=a,bb=0,ab=ba=0{\displaystyle \textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0} for the second algebra,

aa=b,bb=0,ab=ba=0{\displaystyle \textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0} for the third algebra,

Generalization: algebra over a ring

In some areas of mathematics, such as commutative algebra, it is common to consider the more general concept of an algebra over a ring, where a commutative unital ring R replaces the field K. The only part of the definition that changes is that A is assumed to be an R-module (instead of a vector space over K).

Associative algebras over rings

A ringA is always an associative algebra over its center, and over the integers. A classical example of an algebra over its center, is the split-biquaternion algebra, which is isomorphic to H×H{\displaystyle \mathbb {H} \times \mathbb {H} }, the direct product of two quaternion algebras. The center of that ring is R×R{\displaystyle \mathbb {R} \times \mathbb {R} }, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional R{\displaystyle \mathbb {R} }-algebra.

In commutative algebra, if A is a commutative ring, then any unital ring homomorphism R→A{\displaystyle R\to A} defines an R-module structure on A, and this is what is known as the R-algebra structure.[5] So a ring comes with a natural Z{\displaystyle \mathbb {Z} }-module structure, since one can take the unique homomorphism Z→A.{\displaystyle \mathbb {Z} \to A.}[6] On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See the field with one element, for an attempt to give to every ring a structure that behaves like an algebra over a field.